The formula for the arc length of a space curve measures the total distance traveled along the curve between two points in three-dimensional space. This concept is fundamental in calculus for understanding the geometry of paths.
The exact formula for the arc length of a space curve, as provided in the reference "Arc Length Along A Space Curve," is:
For the total arc length (L) over a specific interval:
$L = \int_a^b \sqrt{|\mathbf{v}|}dt$
For the arc length function (s(t)), which measures the length from a starting point $t_0$ to any point t:
$s(t) = \int_{t_0}^t \sqrt{|\mathbf{v}(\tau)|}d\tau$
These equations, which are also applicable to curves in a plane, remain valid for space curves, allowing us to calculate the length of paths in three dimensions.
Understanding the Arc Length Formula
The arc length formula is essentially an integral that sums up infinitesimal segments of the curve. Imagine breaking the curve into tiny straight lines; the formula calculates the length of each tiny segment and adds them all up.
Key Components of the Formula:
The formula relies on the concept of the velocity vector of the curve. If a space curve is described by a position vector $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$, its velocity vector is the derivative of the position vector with respect to time: $\mathbf{v}(t) = \mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$.
Let's break down the symbols used in the formula:
| Symbol | Description |
|---|---|
| L | Represents the total arc length of the curve over the specified interval. |
| s(t) | Denotes the arc length function, which calculates the length of the curve from a specific starting time ($t_0$) up to a variable time ($t$). |
| $\int_a^b$ | Is the definite integral sign, indicating summation from a starting time a to an ending time b. For the arc length function, $\int_{t_0}^t$ signifies integration from the initial time $t_0$ to a variable time $t$. |
| **$\sqrt{ | \mathbf{v} |
| dt or dτ | Represents an infinitesimal change in time. $\tau$ is used as a dummy variable of integration when the upper limit of the integral is also $t$. |
Calculating Arc Length: A Step-by-Step Approach
To calculate the arc length of a space curve, follow these general steps:
- Parametrize the Curve: Ensure the curve is described by a position vector function $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$ over a specified interval $[a, b]$.
- Find the Velocity Vector: Compute the first derivative of the position vector with respect to time, which gives the velocity vector: $\mathbf{v}(t) = \mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$.
- Calculate the Magnitude of the Velocity Vector: Determine the speed of the curve at any given time by finding the magnitude of the velocity vector. This is calculated as:
$||\mathbf{v}(t)|| = \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}$.
(As noted, the reference denotes this as $\sqrt{|\mathbf{v}|}$). - Integrate: Substitute the magnitude of the velocity vector into the arc length formula and perform the definite integral over the interval $[a, b]$:
$L = \int_a^b ||\mathbf{v}(t)|| dt$
or using the reference's notation:
$L = \int_a^b \sqrt{|\mathbf{v}|}dt$.
Extension from Plane Curves to Space Curves
A notable point highlighted by the reference is that the same formula applies to both curves in planes (2D) and space curves (3D). For a plane curve, the position vector would simply be $\mathbf{r}(t) = \langle x(t), y(t) \rangle$, and its magnitude of velocity would be $||\mathbf{v}(t)|| = \sqrt{(x'(t))^2 + (y'(t))^2}$. The underlying principle remains identical: integrating the speed of the curve over time yields its length. This demonstrates the elegance and consistency of vector calculus in describing motion and geometry across different dimensions.
